We show that spatial solitons on either phase- or amplitude-modulated backgrounds can change their direction of motion according to the modulation frequency. A soliton may, therefore, move up or down phase gradients or remain motionless regardless of where it is in relation to the background modulation. The general theory is in good agreement with numerical results in a variety of nonlinear systems.
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Scroggie et al. (2005) studied this question.
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